TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

State similarity preserves input-output data

Statement

Let $d \geq 0$ , $m, p \geq 1$ , and $q \geq 1$ be integers. Suppose a noiseless experiment has matrices $A \in M_{d \times d} (R)$ , $B \in M_{d \times m} (R)$ , and $C \in M_{p \times d} (R)$ , initial state $x_{0} \in R^{d}$ , and common inputs $(u_{k})_{0 \leq k < q - 1}$ . For any invertible $T \in M_{d} (R)$ define $\hat{A} = T^{- 1} A T, \hat{B} = T^{- 1} B, \hat{C} = C T, {\hat{x}}_{0} = T^{- 1} x_{0} .$ The experiment with $(\hat{A}, \hat{B}, \hat{C}, {\hat{x}}_{0})$ and the same inputs has the same output $y_{k}$ at every sample $0 \leq k < q$ . The equality holds for every horizon and every common input sequence.

Statement references

Discrete input-state-output recurrence

Invertible matrices and their inverses

Associativity and identity laws for matrix products

Dahleh, Dahleh, and Verghese, Lectures on Dynamic Systems and Control, §25.3

Lall, State-Space Realization, §9.2

Facts & Assumptions

Given: A noiseless real discrete experiment of horizon $q$ , matrices $A, B, C$ , initial state $x_{0}$ , common inputs $u_{0}, \dots, u_{q - 2}$ , and an invertible matrix $T \in M_{d} (R)$ .

[F1]

Discrete linear input-state-output experiments: the state and output equations are $x_{k + 1} = A x_{k} + B u_{k}$ and $y_{k} = C x_{k}$ at the stated samples.

[F2]

Invertible matrices and the general linear group ${GL}_{n} (F)$ : an invertible $T$ has a two-sided inverse $T^{- 1}$ , so $T T^{- 1} = T^{- 1} T = I_{d}$ .

[F3]

Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication: conformable matrix products are associative and identity matrices act on either side.

Proof

1.1givenF1F2F3algebra

At $k = 0$ , the transformed initial state is ${\hat{x}}_{0} = T^{- 1} x_{0}$ . If ${\hat{x}}_{k} = T^{- 1} x_{k}$ for $k < q - 1$ , then [F1] and the definitions give ${\hat{x}}_{k + 1} = \hat{A} {\hat{x}}_{k} + \hat{B} u_{k} = T^{- 1} A T T^{- 1} x_{k} + T^{- 1} B u_{k} = T^{- 1} (A x_{k} + B u_{k}) = T^{- 1} x_{k + 1}$ ; [F2] cancels $T T^{- 1}$ and [F3] permits regrouping. Induction proves ${\hat{x}}_{k} = T^{- 1} x_{k}$ for $0 \leq k < q$ .

2.1givenF1F2step 1.1algebra∎

For every such $k$ , the transformed output is ${\hat{y}}_{k} = \hat{C} {\hat{x}}_{k} = C T T^{- 1} x_{k} = C x_{k} = y_{k}$ by [F1] and [F2]. Since the horizon, initial state, and inputs were arbitrary, this holds for every common finite input experiment.

Coordinate diagram

The square records the state-coordinate identity $T^{- 1} A = \hat{A} T^{- 1}$ : both paths send $x$ to $T^{- 1} A x$ . The proof above establishes the output identity by induction and does not rely on the diagram.

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Mohammed Dahleh, Munther A. Dahleh, and George Verghese, Lectures on Dynamic Systems and Control, Chapter 25: Minimal State-Space Realization (standard reference, not scraped)
Sanjay Lall, Engr207c: Operator Theory and Linear Systems, State-Space Realization lecture notes (standard reference, not scraped)